Possible. The cause if an upper respiratory infection (sinus / ear). Can be transmitted either from child to patent or vice versa... To determine if bacterial infection vs rhinitis. Length of time, bloody nasal discharge, tender sinuses, headache and even fever are taken into account., .
Sinusitis. Generally sinusitis will present with colored mucous and or pressure over sinus areas some with fever.
Exam and history. Your primary physician or allergist can help with this. Part of it depends on the appearance of the lining of the nose, along with a detailed description of your symptoms. Response to treamtent is another clue - atrophic rhinitis responds best to ipratroprium spray, whereas sinus inflammation is best treated with steroid sprays.
ENT specialist. Would recommend seeing ears, nose, and throat specialist.